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Chapter 1
Introduction
1.1
What is “MDO”?
Multidisciplinary design optimization (MDO) is the application of numerical optimization techniques to the design of engineering systems that involve multiple disciplines. Aircraft are prime examples of multidisciplinary systems, so it is no coincidence that MDO emerged within the aerospace
community.
Before covering MDO, we first need to cover the “O”, i.e. numerical optimization. This consists
in the use of algorithms to minimize or maximize a given function by varying a number of variables.
The problem might or not be subject to constraints.
Since many engineering design problems seek to maximize some measure of performance, it
became obvious that using numerical optimization in design could be extremely useful, hence design
optimization (the “DO” in MDO). Structural design was one of the first applications. A typical
structural design optimization problem is to minimize the weight by varying structural thicknesses
subject to stress constraints.
MDO emerged in the 1980s following the success of the application of numerical optimization
techniques to structural design in the 1970s. Aircraft design was one of the first applications
of MDO because there is much to be gained by the simultaneous consideration of the various
disciplines involved (structures, aerodynamics, propulsion, stability and controls, etc.), which are
tightly coupled. As an example, suppose that you are able to save one unit of weight in the
structure. Because the coupled nature of all the aircraft weight dependencies, and the reduction in
induced drag, the total reduction in the aircraft gross weight will be several times the structural
weight reduction (about 5 for a typical airliner).
In spite of its usefulness, DO and MDO remain underused industry. There are several reasons
for this, one of which is the absence of MDO in undergraduate and graduate curricula. This is
changing, as most top aerospace departments nowadays include at least one graduate level course
on MDO. We have also seen MDO being increasingly used by the major airframes.
The design optimization process follows a similar iterative procedure to that of the conventional
design process, with a few key differences. Figure 1.1 illustrates this comparison. Both approaches
must start with a baseline design derived from the specifications. This baseline design usually
requires some engineering intuition and represents an initial idea. In the conventional design process
this baseline design is analyzed in some way to determine its performance. This could involve
numerical modeling or actual building and testing. The design is then evaluated based on the
results and the designer then decides whether the design is good enough or not. If the answer is no
— which is likely to be the case for at least the first few iterations — the designer will change the
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design based on his or her intuition, experience or trade studies. When the design is satisfactory,
the designer will arrive at the final design.
For more complex engineering systems, there are multiple levels and thus cycles in the design
process. In aircraft design, these would correspond to the preliminary, conceptual and detailed
design stages.
The design optimization process can be pictured using the same flow chart, with modifications
to some of the blocks. Instead of having the option to build a prototype, the analysis step must
be completely numerical and must not involve any input from the designer. The evaluation of the
design is strictly based on numerical values for the objective to be minimized and the constraints
that need to be satisfied. When a rigorous optimization algorithm is used, the decision to finalize the
design is made only when the current design satisfies the necessary optimality conditions that ensure
that no other design “close by” is better. The changes in the design are made automatically by
the optimization algorithm and do not require the intervention of the designer. On the other hand,
the designer must decide in advance which parameters can be changed. In the design optimization
process, it is crucial that the designer formulate the optimization problem well. We will now discuss
the components of this formulation in more detail: the objective function, the constraints, and the
design variables.
Baseline
design
Specifications
Baseline
design
Specifications
Analyze or
experiment
Change
design
No
Evaluate
performance
Is the design
good?
Yes
Final design
Analyze
Change
design
No
Evaluate
objective and
constraints
Is the design
optimal?
Yes
Final design
Figure 1.1: Conventional (left) versus optimal (right) design process
1.2
Terminology and Problem Statement
The formulation of the problem statement is often undervalued. Even experienced designers are
sometimes not clear on what the different components in the optimization problem statement
mean. Common conceptual errors include confusing constraints with objective functions. If we get
the problem formulation wrong, then the solution of that problem could fail or simply return a
mathematical optimum that is not the engineering optimum.
1.2.1
Objective Function
The objective function, f , is the scalar that we want to minimize. This is the convention in
numerical optimization, so if we want to maximize a function, then we simply multiply it by −1
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or invert it. This function thus represents a measure of “badness” that we want to minimize.
Examples include weight in structural design and direct operating cost in aircraft design. To
perform numerical optimization, we require the objective function to be computable for a range of
designs. The objective function can be given as an explicit function, or be the result of a complex
computational procedure, like the drag coefficient given by computational fluid dynamics.
The choice of objective function is crucial: if the wrong function is used, it does not matter how
accurate the computation of the objective is, or how efficient the numerical optimization solves the
problem. The “optimal” design is likely to be obviously non-optimal from the engineering point of
view. A bad choice of objective function is a common mistake in MDO.
Optimization problems are classified with respect to the objective function by determining how
the function depends on the design variables (linear, quadratic or generally nonlinear). In this
course, we will concentrate on the general nonlinear case, although as we will see, there is merit in
analyzing the quadratic case.
One common misconception is that in engineering design we often want to optimize multiple
objectives. As we will se later, it is possible to consider problems with multiple objectives, but this
results in a family of optimum designs with different emphasis on the various objectives. In most
cases, it makes much more sense to convert these “objectives” into constraints. In the end, we can
only make one thing best at the time.
The “Disciplanes”
Is there one aircraft which is the fastest, most efficient, quietest, most inexpensive?
“You can only make one thing best at a time.”
1.2.2
Design Variables
The design variables, x, are the parameters that the optimization algorithm is free to choose in
order to minimize the objective function. The optimization problem formulation allows for upper
and lower bounds for each design variable (also known as side constraints). We distinguish these
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bounds from constraints, since as we will see later, they require a different numerical treatment
when solving an optimization problem.
Design variables must be truly independent variables, i.e., in a given analysis of the design, they
must be input parameters that remain fixed in the analysis cycle. They must not depend on each
other or any other parameter.
Design variables can be broadly classified as quantitative and qualitative. Quantitative variables
can be expressed by a number, which could be continuous or discrete. A real variable will result
in the continuous case if allowed to vary at least within a certain range. If only discrete values
are allowed (e.g., only plates of a certain thickness in a structure are available), then this results
in the discrete case. Integer design variables will invariable result in the discrete case. Qualitative
variables do not represent quantities, but discrete choices (e.g., the choice of aircraft configuration:
monoplane vs. biplane, etc.).
In this course, we will deal exclusively with continuous design variables.
1.2.3
Constraints
The vast majority of practical design optimization problems have constraints. These are functions
of the design variables that we want to restrict in some way. When we restrict a function to being
equal to a fixed quantity, we call this an equality constraint. When the function is required to be
greater or equal to a certain quantity, we have an inequality constraint. The convention we will
use of inequality constraints is greater or equal, so we will have to make sure that less or equal
constraints are multiplied by −1 before solving the problem. Note that some authors consider the
less or equal constraints to be the convention.
As in the case of the objective function, the constraint functions can be linear, quadratic or
nonlinear. We will focus on the nonlinear case, but the linear case will provide the basis for some
of the optimization algorithms.
Inequality constraints can be be active or inactive at the optimum point. If inactive, then the
corresponding constraint could have been left out of the problem with no change in its solution. In
the general case, however, it is difficult to know in advance which constraints are active or not.
Constrained optimization is the subject of Chapter 5.
1.2.4
Optimization Problem Statement
Now that we have defined the objective function, design variables, and constraints, we can formalize
the optimization problem statement as shown below.
minimize
f (x)
by varying x ∈ Rn
subject to ĉj (x) = 0,
j = 1, 2, . . . , m̂
ck (x) ≥ 0,
k = 1, 2, . . . , m
(1.1)
f : objective function, output (e.g. structural weight).
x : vector of design variables, inputs (e.g. aerodynamic shape); bounds can be set on these
variables.
ĉ : vector of equality constraints (e.g. lift); in general these are nonlinear functions of the design
variables.
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Smooth
Discontinuous
Linear
Continuity
Linearity
Nonlinear
Static
Continuous
Dynamic
Quantitative
Discrete
Optimization
Problem
Classification
Time
Design
Variables
Qualitative
Deterministic
Data
Constraints
Convexity
Unconstrained
Stochastic
Constrained
Convex
NonConvex
Figure 1.2: Classification of optimization problems; the course focuses on the types highlighted in
blue.
c : vector of inequality constraints (e.g. structural stresses), may also be nonlinear and implicit.
where f is the objective function, x is the vector of n design variables, ĉ is the vector of m̂ equality
constraints and c is the vector of m inequality constraints.
1.2.5
Classification of Optimization Problems
Optimization problems are classified based on the various characteristics of the objective function,
constraint functions, and design variables. Figure 1.2 shows some of the possibilities.
In this course, we will consider the general nonlinear, non-convex constrained optimization
problems with continuous design variables. However, you should be aware that restricting the
optimization to certain types and using specialized optimization algorithms can result in a dramatic
improvement in the capacity to solve a problem.
1.3
Timeline of Historical Developments in Optimization
300 bc: Euclid considers the minimal distance between a point a line, and proves that a square
has the greatest area among the rectangles with given total length of edges.
200 bc: Zenodorus works on “Dido’s Problem”, which involved finding the figure bounded by a
line that has the maximum area for a given perimeter.
100 bc: Heron proves that light travels between two points through the path with shortest length
when reflecting from a mirror, resulting in an angle of reflection equal to the angle of incidence.
1615: Johannes Kepler finds the optimal dimensions of wine barrel. He also formulated an early
version of the “marriage problem” (a classical application of dynamic programming also
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Steepest
Descent
Gradient
Based
Conjugate
Gradient
QuasiNewton
Optimization
Methods
Grid or
Random
Search
Genetic
Algorithms
Simulated
Annealing
Gradient
Free
Nelder–
Meade
DIRECT
Particle
Swarm
Figure 1.3: Optimization methods for nonlinear problems
known as the “secretary problem”) when he started to look for his second wife. The problem
involved maximizing a utility function based on the balance of virtues and drawbacks of 11
candidates.
1621 W. van Royen Snell discovers the law of refraction. This law follows the more general principle
of least time (or Fermat’s principle), which states that a ray of light going from one point to
another will follow the path that takes the least time.
1646: P. de Fermat shows that the gradient of a function is zero at the extreme point the gradient
of a function.
1695: Isaac Newton solves for the shape of a symmetrical body of revolution that minimizes fluid
drag using calculus of variations.
1696: Johann Bernoulli challenges all the mathematicians in the world to find the path of a body
subject to gravity that minimizes the travel time between two points of different heights
— the brachistochrone problem. Bernoulli already had a solution that he kept secret. Five
mathematicians respond with solutions: Isaac Newton, Jakob Bernoulli (Johann’s brother),
Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l’Hôpital. Newton
reportedly started solving the problem as soon as he received it, did not sleep that night and
took almost 12 hours to solve it, sending back the solution that same day.
1740: L. Euler’s publication begins the research on general theory of calculus of variations.
1746: P. L. Maupertuis proposed the principle of least action, which unifies various laws of physical motion. This is the precursor of the variation principle of stationary action, which uses
calculus of variations and plays a central role in Lagrangian and Hamiltonian classical mechanics.
1784: G. Monge investigates a combinatorial optimization problem known as the transportation
problem.
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1805: Adrien Legendre describes the method of least squares, which was used in the prediction of
asteroid orbits and curve fitting. Frederich Gauss publishes a rigorous mathematical foundation for the method of least squares and claims he used to predict the orbit of the asteroid
Ceres in 1801. Legendre and Gauss engage in a bitter dispute on who first developed the
method.
1815: D. Ricardo publishes the law of diminishing returns for land cultivation.
1847: A. L. Cauchy presents the steepest descent methods, the first gradient-based method.
1857: J. W. Gibbs shows that chemical equilibrium is attained when the energy is a minimum.
1902: Gyula Farkas presents and important lemma that is later used in the proof of the Karush–
Kuhn–Tucker theorem.
1917: H. Hancock publishes the first text book on optimization.
1932: K. Menger presents a general formulation of the traveling salesman problem, one of the most
intensively studied problems in optimization.
1939: William Karush derives the necessary conditions for the inequality constrained problem in
his Masters thesis. Harold Kuhn and Albert Tucker rediscover these conditions an publish
their seminal paper in 1951. These became known as the Karush–Kuhn–Tucker (KKT)
conditions.
1939 Leonid Kantorovich develops a technique to solve linear optimization problems after having
given the task of optimizing production in the Soviet government plywood industry.
1947: George Dantzig publishes the simplex algorithm. Dantzig, who worked for the US Air Force,
reinvented and developed linear programming further to plan expenditures and returns in
order to reduce costs to the army and increase losses to the enemy in World War II. The
algorithm was kept secret until its publication.
1947: John von Neumann develops the theory of duality for linear problems.
1949: The first international conference on optimization, the International Symposium on Mathematical Programming, is held in Chicago.
1951: H. Markowitz presents his portfolio theory that is based on quadratic optimization. He
receives the Nobel memorial prize in economics in 1990.
1954: L. R. Ford and D. R. Fulkerson research network problems, founding the field of combinatorial optimization.
1957: R. Bellman presents the necessary optimality conditions for dynamic programming problems. The Bellman equation was first applied to engineering control theory, and subsequently
became an important principle in the development of economic theory.
1959: Davidon develops the first quasi-Newton method for solving nonlinear optimization problems. Fletcher and Powell publish further developments in 1963.
1960: Zoutendijk presents the methods of feasible directions to generalize the Simplex method for
nonlinear programs. Rosen, Wolfe, and Powell develop similar ideas.
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1963: Wilson invents the sequential quadratic programming method for the first time. Han reinvents it in 1975 and Powell does the same in 1977.
1975: Pironneau publishes the a seminal paper on aerodynamic shape optimization, which first
proposes the use of adjoint methods for sensitivity analysis [12].
1975: John Holland proposed the first genetic algorithm.
1977: Raphael Haftka publishes one of the first multidisciplinary design optimization (MDO) applications, in a paper entitled “Optimization of flexible wing structures subject to strength
and induced drag constraints” [2].
1979: Kachiyan proposes the first polynomial time algorithm for linear problems. The New York
times publishes the front headline “A Soviet Discovery Rocks World of Mathematics”, saying,
“A surprise discovery by an obscure Soviet mathematician has rocked the world of mathematics and computer analysis . . . Apart from its profound theoretical interest, the new discovery
may be applicable in weather prediction, complicated industrial processes, petroleum refining,
t.he scheduling of workers at large factories . . . the theory of secret codes could eventually be
affected by the Russian discovery, and this fact has obvious importance to intelligence agencies everywhere.” In 1975, Kantorovich and T.C. Koopmans get the Nobel memorial price of
economics for their contributions on linear programming.
1984: Narendra Karmarkar starts the age of interior point methods by proposing a more efficient
algorithm for solving linear problems. In a particular application in communications network
optimization, the solution time was reduced from weeks to days, enabling faster business and
policy decisions. Karmarkar’s algorithm stimulated the development of several other interior
point methods, some of which are used in current codes for solving linear programs.
1985: The first conference in MDO, the Multidisciplinary Analysis and Optimization (MA&O)
conference, takes place.
1988: Jameson develops adjoint-based aerodynamic shape optimization for computational fluid
dynamics (CFD).
1995: Kennedy and Eberhart propose the particle swarm optimization algorithm.
1.4
1.4.1
Practical Applications
Airfoil Design
Aerodynamic shape optimization emerged after computational fluid dynamics (CFD) started to be
used more routinely. Once aerodynamicists were able to analyze arbitrary shapes, the natural next
step was to devise algorithms that improved the shape automatically [4, 5]. Initially, 2D problems
where tackled, resulting in airfoil shape optimization. Today, full configurations can be optimized
not only considering aerodynamics [13, 14], but aerodynamics and structures [9].
The airfoil design optimization example we show here are due to Nemec et al. [10]. The solution
shown in Figure 1.4 corresponds to the maximization of the lift-to-drag ratio of the airfoil with
respect to 9 shape variables and angle of attack, subject to airfoil thickness constraints.
Instead of maximizing the lift-to-drag ratio, another possible objective function is the drag
coefficient. In this case, we must impose a lift constraint. Figure 1.5 shows the results for a drag
minimization problem. We can see that the optimized results minimizes the drag at the specified
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64
0
62
-4
Cl/Cd
Cp
-2
-2
58
-1
-3
56
Cl/Cd
Gradient
54
0
52
0
0.2
0.4
0.6
0.8
1
X/C
-4
log(||Gradient||)
60
Initial Design
Final Design
-3
1
-1
-5
50
-6
10
20
30
40
50
Flow Solves and Gradient Evaluations
Figure 1.4: Airfoil shapes, corresponding pressure coefficient distributions, and convergence history
for lift-to-drag maximization case.
condition to a fault and exhibits particularly bad off-design performance. This is a typical example
of the optimizer exploiting a weakness in the problem formulation.
To improve off-design performance, we can include the performance of multiple flight conditions
in the objective function. This is the case for the problem show in in Figure 1.6, in which four
flight conditions where considered.
Figure 1.7 shows the solution of a multiobjective problem with drag and the inverse of lift
coefficient as two objective functions. The solution is a line with an infinite number of designs
called the Pareto front.
1.4.2
Structural Topology Optimization
This example is from recent work by James and Martins [3]. The objective of structural topology
optimization is to find the shape and topology of a structure that has the minimum compliance
(maximum stiffness) for a given loading condition.
Find the shape and topology of a structure that has the minimum compliance (maximum stiffness) for a given loading condition.
1.4.3
Composite Curing Cycle Optimization
This example tackles the problem of optimizing the temperature input in a composite curing cycle [6]. The objective is to maximize the tensile strength of the resulting composite beam.
1.4.4
Aircraft Design with Minimum Environmental Impact
The environmental impact of aircraft has been a popular topic in the last few years. In this example,
Antoine and Kroo [1] show the trade-offs between cost, noise, and greenhouse gas emissions, by
solving a number of multiobjective problems.
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0.024
RAE 2822
Final Design
-1
0.022
0.02
0
Cd
Cp
-0.5
0.018
0.5
RAE 2822
Final Design
1
0
0.25
0.5
0.75
1
X/C
0.016
0.014
0.65
0.7
Mach Number
0.75
Figure 1.5: Airfoil shapes, corresponding pressure coefficient distributions, and convergence history
for single point drag minimization.
0.024
-1
RAE 2822
Final Design
0.022
0.02
0
Cd
Cp
-0.5
0.018
0.5
RAE 2822
Final Design
1
0
0.25
0.5
0.75
1
X/C
0.016
0.014
0.65
0.7
Mach Number
0.75
Figure 1.6: Airfoil shapes, corresponding pressure coefficient distributions, and convergence history
for multipoint drag minimization.
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0.012
0.0115
ωL=0.99
CL=0.549
CD=0.0121
ωL=0.99
ωL=0.7
ωL=0.5
ωL=0.3
ωL=0.2
0.011
CD
ωL=0.1
ωL=0.05
0.0105
ωL=0.01
CL=0.132
CD=0.0104
0.01
0.0095
-0.5
-0.4
-CL
-0.3
ωL=0.01
-0.2
Figure 1.7: Pareto front for drag minimization and lift maximization problem
Figure 1.8: Structural topology optimization process
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Figure 1.9: Optimized topology showing four different levels of refinement
Figure 1.10: Composite beam
Vacuum Bag
Bleeder Cloth
Composite
Bleeder Dam
Tooling
Figure 1.11: Composite beam curing arrangement
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Boundary Temperature
Control Points
500
Temperature [K]
450
400
350
300
0
50
100
150
Time [min]
Figure 1.12: Baseline curing cycle
500
P1
P2
P3
P4
Temperature [K]
450
400
350
300
0
50
100
150
Time [min]
Figure 1.13: Optimized curing cycle
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Figure 1.14: Analysis and optimization framework
Figure 1.15: Pareto fronts of fuel carried, emissions and noise vs. operating cost
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Figure 1.16: Pareto surface of emissions vs. fuel carried vs. noise
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Sunday 1st April, 2012 at 19:48
Figure 1.17: Supersonic business jet configuration
1.4.5
Aerodynamic Design of a Natural Laminar Flow Supersonic Business Jet
The goal of this project was to aid the design of a natural laminar flow wing for a supersonic business
jet, a concept that is being developed by Aerion Corporation and Desktop Aeronautics [8].
In this work, a CFD Euler code was combined with a boundary-layer solver to compute the flow
on a wing-body. The fuselage spoils the laminar flow that can normally be maintained on a thin,
low sweep wing in supersonic flow. The goal is to reshape the fuselage at the wing-body junction to
maximize the extent of laminar flow on the wing. Three design variables were used initially, with
quadratic response surfaces and a trust region update algorithm.
The baseline design, whose solution is shown in Figure 1.19, is a Sears–Haack body with wing.
This results in early transition (the white areas in the boundary-layer solution). N ∗ is the measure
of laminar instability, with 1.0 (white) being the prediction of transition. The flow is then turbulent
from the first occurrence of N ∗ = 1 to the trailing edge irrespective of further values of N ∗ .
With only three design variables (the crosses on the fuselage outline that sit on the wing) and
two iterations (not even near a converged optimization) the improvement is dramatic.
With five design variables, and a few more trust-region update cycles, a better solution is found.
The boundary layer is much farther from transition to turbulent flow as can be seen by comparing the green and yellow colors on this wing with the red and violet colors in the three variables
case. Also notice how subtle the reshaping of the fuselage is. This is typical of aerodynamic shape
optimization: small changes in shape have a big impact in the design.
1.4.6
Aerostructural Design of a Supersonic Business Jet
See Martins et al. [9] for details.
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Figure 1.18: Aerodynamic analysis showing the boundary-layer solution superimposed on the Euler
pressures
Figure 1.19: Baseline design. The colors on the wing show a measure of laminar instability: White
areas represent transition to turbulent boundary layer.
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Sunday 1st April, 2012 at 19:48
Figure 1.20: Fuselage shape optimization with three shape variables. From the nose at left, to
the tail at right, this is the radius of the original (blue) and re-designed (red) fuselage after two
iterations.
Figure 1.21: Laminar instability for fuselage optimized with three design variables
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Figure 1.22: Fuselage shape optimization with five shape variables.
Figure 1.23: Laminar instability for fuselage optimized with 5 design variables
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Flow
Surface
pressures
Geometry
Mesh
displacements
Nodal forces
Nodal
displacements
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Structure
• Aerodynamics: a parallel, multiblock
Euler flow solver.
• Structures: detailed finite element model
with plates and trusses.
• Coupling: high-fidelity, consistent and
conservative.
• Geometry: centralized database for
exchanges (jig shape, pressure
distributions, displacements)
• Coupled-adjoint sensitivity analysis:
aerodynamic and structural design
variables.
Supersonic business jet specification:
• Natural laminar flow
Sunday 1st April, 2012 at 19:48
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WYX"Z![\]X5^`_acb"d7e+f#ghji"[klf4[m\ d7e
n oqpYr4s5t+uvwpyx"zz{x}|~€u
LNMPO5Q5RSTU"V
ƒ‚ „ †ˆ‡
&')(+*,.-0/21",435*5-67(8(:9;=<?>@(
ACBED5F5GHI"J"K
‰‹Š"Œ7Ž".Ž‘“’•”—– ˜.™ š
!"#"$% Figure 1.24: Baseline configuration and design variables
• Cruise at Mach = 1.5
• Range = 5,300nm
• 1 count of drag = 310 lbs of weight
Objective function to be minimized:
I = αCD + βW
(1.2)
where CD is that of the cruise condition. The structural constraint is an aggregation of the stress
constraints at a maneuver condition, i.e.,
KS(σm ) ≥ 0
(1.3)
The design variables are external shape and internal structural sizes.
1.4.7
Aerostructural Shape Optimization of Wind Turbine Blades Considering SiteSpecific Winds
See Kenway and Martins [7] for details.
• What effect does a location’s particular wind distribution have on optimum design?
• Two sites are chosen to represent the two differing environments:
• Direct simulation of the total mechanical output
Objective: Minimize the cost of energy, COE =
Design Variables:
C
AEP
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0.2
80
9000
0.1
0
75
70
Weight (lbs)
-0.2
-0.3
7000
-0.4
-0.5
6000
-0.6
-0.7
5000
65
-0.8
KS
-0.9
Drag
4000
-1
Weight
60
KS
Drag (counts)
-0.1
8000
-1.1
-1.2
3000
0
10
20
30
Major iteration number
40
50
Figure 1.25: Optimization history
LNM
V !W XZY\[]
O
O
PQ PPRSUTT
^_ ^U`aR bdc#e
!#"%$'&)( +* ,$-"./
02143 5 -6 7.897:7<;>=?8@7A7CBEDCF38!GIH8!=KJ
Figure 1.26: Optimized design
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Time Fraction
0.2
0.15
St. Lawrence
UTIAS
0.1
0.05
0
0
10
20
Wind Speed (m/s)
30
Figure 1.27: Wind speed distributions (left) and the Wes5 Tulipo wind turbine (right).
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Design Variable
Chord
Twists
Wspar
tspar
tfoil
Ω
Count
4
4
4
4
3
varies (12)
Lower Limit
.05 m
-75 deg
4%
0.3 mm
6%
7.5 rad/s
Upper Limit
.40 m
75 deg
30%
10mm
20%
14.7 rad/s
Constraints:
Constraint
Stress
Spar Mass
Surface Area
Power
Geometry
7m/s
Minimum
0.5mm
Maximum
40MPa
3.7kg
0.83m2
5000 W
-
12m/s
33
AA222: MDO
Location
St. Lawrence
UTIAS
P̄init (W)
1566.1
660.2
7 m/s
P̄opt (W)
1984.5
853.3
P̄other−opt (W)
1905.1
826.0
Sunday 1st April, 2012 at 19:48
Site-specific increase
4.17%
3.31%
12 m/s
AA222: MDO
1.4.8
34
Sunday 1st April, 2012 at 19:48
MDO of Supersonic Low-boom Designs
The goal in low-boom design is to reduce the strength of the sonic boom sufficiently to permit
supersonic flight overland. The idea is to make subtle changes to the shape of the aircraft that
lead to changes in the sonic boom at ground level. This is a very challenging MDO problem. For
example, the shape has a direct impact on both the aerodynamic performance and the sonic-boom.
In addition, the strength of the sonic boom is related to the weight of the aircraft and length of
the aircraft.
One idea for low-boom design is to find (somehow) a target equivalent-area distribution that
produces an acceptable ground level boom. Subsequently, we can use CFD-based inverse design
to shape an aircraft to match the desired target equivalent area. This is the approach taken in
Palacios [11].
Propagation of the sonic boom from the aircraft near field to the ground.
Early geometry for the Nasa/Lockheed N+2 configuration.
35
AA222: MDO
Sunday 1st April, 2012 at 19:48
Pressure field (lower half) and equivalent-area shape sensitivity fields.
Bibliography
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Journal of Aircraft, 41(4):790–797, 2004.
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drag constraints. AIAA Journal, 14(8):1106–1977, 1977.
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for multiple load cases using a dynamic aggregation technique. Engineering Optimization, 41
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AA222: MDO
36
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[11] Francisco Palacios, Juan Alonso, Michael Colonno, Jason Hicken, and Trent Lukaczyk. Adjointbased method for supersonic aircraft design using equivalent area distributions. In 50th AIAA
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